To solve the given equation, we can first simplify it by distributing the [tex]\sqrt{2}[/tex] to both terms:

[tex]x^2 - (\sqrt{2} + 1)x\sqrt{2} = 0[/tex]
[tex]x^2 - \sqrt{2} \cdot x - x \cdot \sqrt{2} = 0[/tex]
[tex]x^2 - 2x = 0[/tex]

Now we have a simpler quadratic equation [tex]x^2 - 2x = 0[/tex] where we can factor out an x:

[tex]x(x - 2) = 0[/tex]

Setting each factor to zero to solve for x:

[tex]x = 0[/tex] or [tex]x - 2 = 0[/tex]

Therefore, the solutions to the equation [tex]x^2 - (\sqrt{2} + 1)x\sqrt{2} = 0[/tex] are [tex]x = 0[/tex] and [tex]x = 2[/tex].